{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(P, \\le)$ be a finite [[partially ordered set|0QV2JS9RJ3]].\nLet $h$ be the largest number of elements of a [[chain|0S8FE7ZMFM]] in $P$, that is, the [[height|MPR6MH4YSQ]] of $P$.\nThen $P$ is the union of $h$ pairwise disjoint [[antichains|03HF498P1J]], and $P$ is not the union of fewer than $h$ antichains.\n\n## Remarks\n\nA chain meets an antichain in at most one element, so a chain with $h$ elements needs $h$ different antichains to cover it.\nFor the other half, let $f(x)$ be the largest number of elements of a chain whose greatest element is $x$.\nIf $x \u003c y$, then a chain ending at $x$ extends by $y$, so $f(x) \u003c f(y)$.\nThus each level set $\\set{x \\in P : f(x) = k}$ is an antichain, and the values of $f$ are $1, \\ldots, h$.\nExample: in the divisors of $12$ ordered by divisibility, the chain $1, 2, 4, 12$ has $4$ elements, and the levels $\\set{1}$, $\\set{2, 3}$, $\\set{4, 6}$, $\\set{12}$ are $4$ antichains that cover the set.\nThe same proof works for an infinite partially ordered set of finite height.\nThe theorem is the dual of [[Dilworth's theorem|GT224XDA4E]], which exchanges the roles of chains and antichains, but its proof is much easier.\n\n## Explanations\n\n### L1 human\n\nThink of the elements as tasks, where $x \u003c y$ means that $x$ must be done before $y$.\nAn antichain is a set of tasks that can be done in the same round.\nThe theorem says that the least number of rounds equals the length of the longest sequence of tasks that must be done one after the other.\n\n### L2 human\n\nThe proof gives an explicit partition: put each element into the level given by the longest chain that ends at it.\nThis is the layering used in scheduling and in drawing directed acyclic graphs.\nIn the language of graphs, the comparability graph of a poset has chromatic number equal to its clique number.\nThe same holds for each induced subgraph, so comparability graphs are perfect.\nTogether with Dilworth's theorem it gives the [[Erdős–Szekeres theorem|0SNJM8HSD3]] on monotone subsequences.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "0S8FE7ZMFM",
      "03HF498P1J",
      "MPR6MH4YSQ"
    ],
    "kind": "statement",
    "msc": [
      "06A07"
    ],
    "name": "Mirsky's theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q6874717",
      "wikipedia": "Mirsky's theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Mirsky's_theorem"
      }
    ],
    "tag": "5VPXR3RDF1",
    "uses": [
      "0QV2JS9RJ3",
      "0S8FE7ZMFM",
      "03HF498P1J",
      "MPR6MH4YSQ"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/5VPXR3RDF1/",
  "links": [
    {
      "field": "about",
      "target": "0S8FE7ZMFM"
    },
    {
      "field": "about",
      "target": "03HF498P1J"
    },
    {
      "field": "about",
      "target": "MPR6MH4YSQ"
    },
    {
      "field": "uses",
      "target": "0QV2JS9RJ3"
    },
    {
      "field": "uses",
      "target": "0S8FE7ZMFM"
    },
    {
      "field": "uses",
      "target": "03HF498P1J"
    },
    {
      "field": "uses",
      "target": "MPR6MH4YSQ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5VPXR3RDF1.md",
  "msc": [
    "06A07"
  ],
  "name": "Mirsky's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5VP/5VPXR3RDF1.md",
  "status": null,
  "tag": "5VPXR3RDF1",
  "uri": "m2://5VPXR3RDF1",
  "url": "https://tamnd.github.io/math2/wiki/mirskys-theorem/",
  "writing": {
    "tag": "5VPXR3RDF1",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}